Monotonicity in the damped pendulum type equations

Abstract

In this paper we investigate the monotonicity in the pendulum type equations with position dependent damping. We show that the system is strongly monotone under the overdamped condition. In the underdamped case, the Poincare map PT is strongly monotone in a forward invariant region provided the average of the external force is large enough. Combining the strong monotonicity with the dissipation property we show that the Poincare map has in the cylindrical phase space an invariant circle, on which PT is actually an orientation preserving circle homeomorphism. A series of consequences has then been obtained, including the existence and uniqueness of the average velocity. Furthermore, we discuss the smoothness of this invariant curve and the ergodicity of PT on this curve.